# Cartesian Product is Small

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## Theorem

Let $a$ and $b$ be small classes.

Then their Cartesian product $a \times b$ is small:

- $\map {\mathscr M} {a \times b}$

## Proof

\(\ds a \times b\) | \(=\) | \(\ds \set {\tuple {x, y}: x \in a \land y \in b}\) | Definition of Cartesian Product | |||||||||||

\(\ds \) | \(=\) | \(\ds \set {\tuple {x, y}: \set x \subseteq a \land \set y \subseteq b}\) | Singleton of Element is Subset | |||||||||||

\(\ds \) | \(\subseteq\) | \(\ds \set {\tuple {x, y}: \set {x, y} \subseteq a \cup b}\) | Set Union Preserves Subsets | |||||||||||

\(\ds \) | \(\subseteq\) | \(\ds \set {\tuple {x, y}: \tuple {x, y} \subseteq \powerset {a \cup b} }\) | Power Set of Subset and Subset Relation is Transitive |

So by definition of power set:

- $a \times b \subseteq \powerset {\powerset {a \cup b} }$

By Union of Small Classes is Small, $a \cup b$ is small.

By the Axiom of Powers, $\powerset {\powerset {a \cup b} }$ is small.

By Axiom of Subsets Equivalents, $a \times b$ is small.

$\blacksquare$

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 6.2$